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<li class="toctree-l1 current"><a class="current reference internal" href="#">Theory and Methods</a><ul>
<li class="toctree-l2"><a class="reference internal" href="#point-kernel-method">Point-Kernel Method</a></li>
<li class="toctree-l2"><a class="reference internal" href="#uncollided-photon-exposure-rate">Uncollided Photon Exposure Rate</a></li>
<li class="toctree-l2"><a class="reference internal" href="#buildup-factors">Buildup Factors</a></li>
<li class="toctree-l2"><a class="reference internal" href="#quadrature">Quadrature</a></li>
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  <section id="theory-and-methods">
<h1>Theory and Methods<a class="headerlink" href="#theory-and-methods" title="Permalink to this heading"></a></h1>
<section id="point-kernel-method">
<h2>Point-Kernel Method<a class="headerlink" href="#point-kernel-method" title="Permalink to this heading"></a></h2>
<p>ZapMeNot calculates the uncollided photon flux using the point-kernel equation:</p>
<p><span class="math notranslate nohighlight">\(\phi(r) = \frac{S}{4\pi r^{2}}e^{-\mu r}\)</span></p>
<p>where</p>
<p><span class="math notranslate nohighlight">\(\phi(r)\)</span> = space dependent uncollided photon flux</p>
<p><span class="math notranslate nohighlight">\(S\)</span> = point isotropic source of mono-energetic photons</p>
<p><span class="math notranslate nohighlight">\(r\)</span> = distance from source location to flux location</p>
<p><span class="math notranslate nohighlight">\(\mu\)</span> = total attenuation coefficient for medium</p>
<p>When multiple materials exist in the model, sum <span class="math notranslate nohighlight">\(\mu_{i} r_{i}\)</span> over all of
the material regions between the source and the flux location.</p>
</section>
<section id="uncollided-photon-exposure-rate">
<h2>Uncollided Photon Exposure Rate<a class="headerlink" href="#uncollided-photon-exposure-rate" title="Permalink to this heading"></a></h2>
<p>The exposure rate from uncollided photons can be calculated as:</p>
<p><span class="math notranslate nohighlight">\(D_{u}(r) = \phi(r) \Re\)</span></p>
<p>and</p>
<p><span class="math notranslate nohighlight">\(\Re = 1.835\cdot 10^{-8} E\left ( \frac{\mu _{en}\left ( E \right )}{\rho } \right )_{air}\)</span></p>
<p>where <span class="math notranslate nohighlight">\(\Re\)</span> has units of R cm<sup>2</sup>,</p>
<p><span class="math notranslate nohighlight">\(E\)</span> is the photon energy in MeV, and</p>
<p><span class="math notranslate nohighlight">\(\mu _{en}/\rho\)</span> is the mass energy deposition coefficient of air in cm<sup>2</sup>/g.</p>
</section>
<section id="buildup-factors">
<h2>Buildup Factors<a class="headerlink" href="#buildup-factors" title="Permalink to this heading"></a></h2>
<p>The buildup factor <span class="math notranslate nohighlight">\(B(r)\)</span> is defined as the ratio of the dose from all photons (collided and uncollided) to
the dose from uncollided photons.  The buildup factor will vary with distance from the source location and the material
traversed.  As such the total exposure from all photons can be calculated as:</p>
<p><span class="math notranslate nohighlight">\(D_{T}(r) = D_{u}(r) B(\mu r)\)</span></p>
<p>where</p>
<p><span class="math notranslate nohighlight">\(D_{T}(r)\)</span> = the space dependent dose from collided and uncollided photons</p>
<p><span class="math notranslate nohighlight">\(B(\mu r)\)</span> = the space dependent buildup factor</p>
<p>Buildup factors are generally derived from analytical results and are approximated by mathematical functions.  The
form of approximation used in ZapMeNot is the <em>geometric progression</em> (GP) form.  The GP coefficients for calculating
the buildup factor as provided in ANSI/ANS-6.4.3-1991 “Gamma-Ray Attenuation coefficients
and Buildup Factors for Engineering Materials.”</p>
<p>Note that the buildup factors in GP form as provided in ANSI/ANS-6.4.3 are only valid for distances
up to 40 mean free paths.  This is generally not a limitation, as the uncollided flux at some energy E
traversing 40 mean free paths has been reduced by a factor of at least <span class="math notranslate nohighlight">\(10^{-13}\)</span>.
However, extrapolation of buildup factors out to 60 mean free paths has been implemented
in ZapMeNot based on methods described in <a class="reference external" href="https://www.kns.org/files/pre_paper/1/13F-06A-2A-김경오.pdf">“Evaluation of Geometric Progression (GP) Buildup Factors Using MCNPX 2.7.0”</a> by Kim, et al., 2013.</p>
</section>
<section id="quadrature">
<h2>Quadrature<a class="headerlink" href="#quadrature" title="Permalink to this heading"></a></h2>
<p>The point kernel method can be applied directly for point sources.  However, distributed
sources required an additional level of modeling.  The point kernel method is extended
to distributed sources using the method of numerical quadrature.  Put simply, the
source volume or area is subdivided into a spatial quadrature and the source within
a spatial mesh is represented as a point source.  Summing the dose responses
resulting from the point sources returns the dose response resulting
from the distributed source. The spatial mesh is not required to be uniform.  The quadrature
weights are normally computed as follows:</p>
<p><span class="math notranslate nohighlight">\(W_{k}=\frac{V_{k}}{\sum_{k=1}^{N}V_{k}}\)</span></p>
<p>where</p>
<p><span class="math notranslate nohighlight">\(W_{k}\)</span> = the source mesh weight</p>
<p><span class="math notranslate nohighlight">\(V_{k}\)</span> = the volume or area of the k<sup>th</sup> mesh</p>
</section>
<section id="multiple-photon-energies">
<h2>Multiple Photon Energies<a class="headerlink" href="#multiple-photon-energies" title="Permalink to this heading"></a></h2>
<p>A typical photon source will emit photons with a number of discrete photon energies,
each with a unique intensity.  These photon sources are modeled by performing a numerical quadrature
analysis for each photon energy with an associated energy-dependent buildup factor.  Photon energies
are limited to a range of 15 keV to 15 MeV to ensure applicability of the attenuation coefficients
and buildup factors presented in ANSI/ANS-6.4.3-1991 “Gamma-Ray Attenuation coefficients
and Buildup Factors for Engineering Materials.”</p>
<p>Frequently the number of discrete photon energies can be in the 100’s.  However, many of these photons energies
are low enough that the photon contribution to dose is small.  Many of the photon energies may also have a very low
intensity due to a small decay branching fraction.  The evaluation is therefore simplified by limiting the number
of discrete photon energies to 40.  A 40 photon energy-group structure is used in the event that a source has more than
40 discrete photon energies.  The energy-group structure is linearly distributed between the highest and lowest photon energies
in the source.  The photon intensity of each energy group is determined such that the energy flux
of photons within the group is preserved.</p>
</section>
</section>


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